Number dictionary · Exact arithmetic
684: factors and number facts
six hundred eighty-four is an even number and is composite. Its 18 positive divisors form 9 factor pairs.
Number JD Editorial Team · Updated 2026-10-08
- In words
- six hundred eighty-four
- Roman numeral
- DCLXXXIV
- Prime factorization
- 2^2 × 3^2 × 19
- Digital root
- 9
All factors of 684
A positive factor divides 684 with no remainder. In ascending order, the complete list is:
1, 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684
Factor pairs
- 1 × 684 = 684
- 2 × 342 = 684
- 3 × 228 = 684
- 4 × 171 = 684
- 6 × 114 = 684
- 9 × 76 = 684
- 12 × 57 = 684
- 18 × 38 = 684
- 19 × 36 = 684
Each pair describes an arrangement of 684 items into equal rows. The pairs have different row and column counts; 684 cannot make a square array of whole items.
Divisor totals
- Positive divisors
- 18
- Sum of all divisors
- 1820
- Sum excluding 684
- 1136
- Classification
- abundant
684 is abundant because its proper divisors add to 1136, which is greater than 684.
Prime factorization of 684
684 = 2^2 × 3^2 × 19
Repeated division, step by step
- 684 ÷ 2 = 342
- 342 ÷ 2 = 171
- 171 ÷ 3 = 57
- 57 ÷ 3 = 19
- 19 ÷ 19 = 1
Multiply the prime divisors in these steps to reconstruct 684. An exponent records how many times the same prime occurs. The divisor-count calculation is (2 + 1) × (2 + 1) × (1 + 1) = 18.
The preceding prime is 683. The next larger prime is 691. The factor list above explains why 684 is composite.
Learn the factorization method684 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | six hundred eighty-four |
| Roman numeral | DCLXXXIV |
| Binary (base 2) | 1010101100 |
| Octal (base 8) | 1254 |
| Decimal (base 10) | 684 |
| Hexadecimal (base 16) | 2AC |
Check the binary conversion
Read 1010101100 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
512 + 128 + 32 + 8 + 4 = 684
Check the hexadecimal conversion
2 × 16^2 + 10 × 16^1 + 12 × 16^0 = 684
Hexadecimal uses A through F for digit values 10 through 15. These notations represent the same quantity; changing the base does not change the integer.
Square, cube and square root of 684
- 684 squared
- 684 × 684 = 467,856
- 684 cubed
- 684 × 684 × 684 = 320,013,504
- Square root
- ≈ 26.153394
- Simplified radical
- 6√19
684 lies between 676 (26²) and 729 (27²), so it is not a perfect square. It is not a perfect cube.
684 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 684.
Distinguish a square from a square rootMultiplication table for 684
These are the first twelve positive multiples. Unlike the finite factor list, the sequence of multiples continues without an upper limit.
Divisibility checks
| Divide by | Whole quotient | Remainder | Divides exactly? |
|---|---|---|---|
| 2 | 342 | 0 | Yes |
| 3 | 228 | 0 | Yes |
| 4 | 171 | 0 | Yes |
| 5 | 136 | 4 | No |
| 6 | 114 | 0 | Yes |
| 7 | 97 | 5 | No |
| 8 | 85 | 4 | No |
| 9 | 76 | 0 | Yes |
| 10 | 68 | 4 | No |
| 11 | 62 | 2 | No |
| 12 | 57 | 0 | Yes |
For example, 684 = 7 × 97 + 5. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 684
684 contains 3 decimal digits. Its digit sum is 6 + 8 + 4 = 18.
- 684: 6 + 8 + 4 = 18
- 18: 1 + 8 = 9
The digital root is 9. Reversing its digits gives 486; it is not a decimal palindrome. Digit patterns depend on the notation being used.
Digital-root arithmetic always finishes at a single digit for a positive integer. Some numerology methods stop at 11, 22 or 33 instead; that is a separate interpretive convention.
Fractions and a worked comparison
As a fraction of 100, 684/100 simplifies to 171/25 by dividing both terms by their greatest common divisor, 4. Its decimal value is 6.84. Because the numerator exceeds the denominator, this value is greater than one.
The greatest common divisor of 684 and 12 is 12. Their least common multiple is 684. The identity GCD × LCM = 684 × 12 = 8208 checks both results.
Work through GCF and LCMQuestions about 684
What are the factors of 684?
The positive factors are 1, 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684. There are 18 factors, including 1 and 684.
Is 684 a prime number?
No. 684 is composite: 2^2 × 3^2 × 19 = 684.
How do you write 684 in Roman numerals?
684 is DCLXXXIV in standard subtractive Roman numeral notation.
What is the binary representation of 684?
684 in decimal is 1010101100 in base 2. Adding its active place values gives 512 + 128 + 32 + 8 + 4 = 684.
Methods and reference definitions
Number JD calculates each result from the integer on this page. Positive factors are enumerated in pairs; repeated division gives the prime factorization; place values verify base conversions. Decimal square roots are rounded to six places when irrational.
NUMBER JOURNAL / JD
A reflective note on 684
Meaning varies by tradition and personal experience. One sighting is not evidence of what will happen next.
The digits of 684 add to 18. That arithmetic result is separate from the cultural symbolism of the pattern. If this number matters to you, write down what you noticed and choose one grounded action rather than treating the sighting as a prediction.